Minimum Chords in Minkowski Planes

Abstract.

We extend the notion of minimum chords of circles from Euclidean geometry to the geometry of real two dimensional Banach spaces, also called Minkowski planes. In terms of unit discs, we first discuss the existence and uniqueness of minimum chords with respect to a given point of a Minkowskian disc. Then we study properties of the set of points of a Minkowskian disc whose minimum chords have lengths not less than a given positive number. Furthermore, the relations between the norms of such points and the lengths of corresponding minimum chords are investigated. New characterizations of the Euclidean plane are also derived.